Three dimensional deep wavelet scattering for quantum energy interpolation

物理量通常被计算为复杂数学方程系统的解，这可能需要对复杂的物理状态进行大量计算。分子能量的量子化学计算就是这样一个例子。事实上，计算分子的能量，给定其原子核的电荷和位置，是计算化学的核心问题，在分子动力学，材料科学和药物发现中具有重要应用。机器学习算法不模拟物理系统，而是通过从已知示例的训练集学习来估计解决方案。然而，这样的学习算法可能需要在系统维度上呈指数的大量示例，因此是难以处理的;这种现象被称为“维度灾难”。这一建议将发展一种基于“散射变换“的分子能量估算新方法。“散射变换估计分子能量，并通过利用多尺度，多级结构，利用物理不变量，规避维数灾难。由此产生的算法有可能显着加快计算高度准确的分子能量估计，导致大规模的原子模拟，大大提高了精度，速度和适应性，从而改变了多尺度建模的范式。此外，PI还将指导一名本科生，并培训一名研究生，从而为将核心思想传播给更广泛的受众创造条件。散射变换具有深度卷积网络的结构，但由迭代小波变换和复模算子组成。这样的网络已经在计算机视觉中用于二维图像的分析和分类以及涉及一维信号的音频任务。多尺度三维散射变换网络在实践（多尺度3D）和设计（量子化学）方面都是新颖的，并且有机会影响这些类型的架构向前发展。系统的方法将在几个方面攻击主要目标：（1）开发具有适当对称性和稳定性的高效3D滤波器;（2）对分子能量泛函的各种组分的散射回归算法进行严格的误差分析;（3）通过快速多极方法的可证明关系更深入地理解散射网络。用于实现这些目标的方法将包括：（一）小波滤波器设计和有效的信号处理算法;（二）利用Littlewood-Paley理论结合多项式（Taylor）近似理论;（三）多尺度分析;（四）验证方法的数值实验。通过将深度学习架构与物理化学严格联系起来，该提案中的研究将在数据科学和科学计算的界面上进行，以实现双方的共同利益。

Physical quantities are often computed as solutions to a system of complex mathematical equations, which may require huge computations for intricate physical states. Quantum chemistry calculations of molecular energies is such an example. Indeed, computing the energy of a molecule, given the charges and positions of its nuclei, is a central issue in computational chemistry with important applications in molecular dynamics, materials science, and drug discovery. Machine learning algorithms do not simulate the physical system but estimate solutions by learning from a training set of known examples. However, such learning algorithms may require a number of examples that is exponential in the system dimension, and are thus intractable; this phenomena is referred to as the "curse of dimensionality." This proposal will develop a novel approach for the estimation of molecular energies based on the "scattering transform." The scattering transform estimates molecular energies, and circumvents the curse of dimensionality, by utilizing a multiscale, multilevel architecture that takes advantage of physical invariants. The resulting algorithms have the potential to significantly speed up the computation of highly accurate molecular energy estimates, leading to large scale atomistic simulations with greatly improved accuracy, speed, and adaptability, thus shifting the paradigm of multiscale modeling. The PI will additionally mentor an undergraduate student and train a graduate student in this field, thus setting up the potential for dissemination of the core ideas to a broader audience.The scattering transform has the structure of a deep convolutional network, but is composed of iterated wavelet transforms and complex modulus operators. Such networks have been used in computer vision for the analysis and classification of two dimensional images and audio tasks involving one dimensional signals. A multiscale three dimensional scattering transform network is novel both in practice (multiscale 3D) and design (for quantum chemistry), and has the chance to influence these types of architectures moving forward. A systematic approach will attack the primary object on several fronts: (1) development of efficient 3D filters with the appropriate symmetry and stability properties; (2) rigorous error analysis of the scattering regression algorithm for various components of the molecular energy functional; (3) deeper understanding of the scattering network via provable relations with fast multipole methods. The methods used to carry out these objectives will include: (i) wavelet filter design and efficient signal processing algorithms; (ii) utilization of Littlewood-Paley Theory in conjunction with polynomial (Taylor) approximation theory; (iii) multiscale analysis; (iv) numerical experiments to validate methods. By rigorously linking deep learning architectures with physical chemistry, the research in this proposal will take place at the interface of data science and scientific computation, for the mutual gain of both.